FORMATION AND
EVOLUTION OF GALAXIES
The Virial theorem
Lecture 16
• Galaxy Formation
• Virial theorem
• Jeans instability for forming stars
• Jeans length and mass
V
Somak Raychaudhury
The Virial Theorem
2�KE� + �P E� = 0
For a spherical cloud of temperature T,
• Assuming constant density
• ignoring magnetic fields
• neglecting rotation
Cloud’s internal kinetic energy=
Cloud’s total potential kinetic energy=
Condition for
Collapse
2�KE� < �P E� or
�KE� =
3
N kT
2
�P E� = −
3 GMc2
5 Rc
3N kT <
3 GMc2
5 Rc
1
Carroll & Ostlie 12.12
Condition for collapse
The radius of the cloud
3 GMc2
3N kT <
5 Rc
The Jeans mass
Interstellar Medium (ISM) - HI regions
.
§  Electrons have charge and spin, and so have a magnetic dipole
moment.
§  Pauli Exclusion Principle:. For a pair of electrons, if their spins are
aligned, they have slightly more energy, than if they had opposite
spin
§  Energy difference is small ~6x10-6 eV, which is ~21 cm (radio
waves).
From radio surveys of our galaxy we find
lots of HI gas: about 1 solar mass of
gas for every 10 solar masses of stars.
Interstellar
Medium (ISM) HII regions
•
•
•
•
Radio
IR
•
HST: Keyhole nebula
HII regions are found near hot stars.
UV photons from nearby stars ionize H atoms.
When electrons and protons recombine, they generally emit
Balmer lines.
Also observe lines from other atoms (e.g., oxygen, sulphur,
silicon, carbon, …).
HII regions are associated with active star formation.
2
Cloud Collapse and
Star/Planet
Formation
Gravitational Collapse in the ISM
•
MJ =
•
•
•
Diffuse HI Cloud
H2 Cloud Core
T
50 K
10 K
ρ
5× 108 m-3
1010 m-3
MJ
1500 M¤
10 M¤
Mc
1-100 M¤
10-1000 M¤
The Jeans Mass
�
5kT
GµmH
�3/2 �
3
4πρ0
�1/2
We know from the Jeans Criterion that if Mc>MJ collapse occurs.
Substituting the values from the table into
gives:
–  Diffuse HI cloud: MJ ~ 1500 MSun => stable as Mc<MJ.
–  Molecular cloud core: MJ ~ 10 Msun => unstable as Mc>MJ.
So deep inside molecular clouds the cores are collapsing to form stars.
Jeans cloud collapse equations
describe the conditions required
for an ISM cloud to collapse.
•
As a cloud collapses, central
temperature increases.
•
This is accompanied by
spinning-up of the central star
(to conserve AM).
•
If the cloud is rotating, the disk
also flattens into an oblate
spheroid.
Problem of star formation efficiency
Time-scale for collapse
•
The collapse time-scale tff when M >MJ is given by the time a mass element
at the cloud surface needs to reach the centre.
•
In free-fall, an mass element is subject to acceleration
•
By approximating R using R3 ~ M/ρ => tff ≈ (G ρ)-1/2
•
Higher density at cloud centre = > faster collapse.
•
For typical molecular cloud, tff ~ 103 years (ie very short).
g=
•
Gas in the galaxy should be wildly gravitationally
unstable. It should convert all its mass into stars on a
free-fall time scale:
GM
R2
t ff =
3"
3.4 $1010
=
yr
32G #
n
For interstellar medium (ISM) in our galaxy:
n " 2 #10 5 m-3
t ff = 8 "10 6 yr
!
!
Total amount of molecular gas in the Galaxy:
~ 2 "10 9 M sun
~!250 M sun / year
!
~ 3 M sun / year
Observed star formation rate:
!
Something slows star formation down...
Expected star formation rate:
!
!
3
Magnetic field support
Magnetic field support
Consider an initially stable cloud. We now compress it. The
density thereby increases, but the mass of the cloud stays
constant.
In presence of B-field, the stability analysis changes.
Magnetic fields can provide support against gravity.
Jeans mass decreases:
MJ "
1
#
If no magnetic fields: there will come a time when M>MJ
and the cloud will collapse.
Replace Jeans mass with critical mass, defined as:
M cr = 0.12
$ B '$ R '
"M
# 10 3 M sun &
)&
)
1/2
G
% 3nT (% 2pc (
2
nT is
nanoTesla!
But Mcr stays constant (the magnetic
! flux will be frozen in
the cloud)
So if B-field is strong enough to support a cloud, no
compression will cause it to collapse.
!
4